Serial irradiation of a substrate by multiple radiation sources

ABSTRACT

A method for configuring J electromagnetic radiation sources (J≧2) to serially irradiate a substrate. Each source has a different function of wavelength and angular distribution of emitted radiation. The substrate includes a base layer and I stacks (I≧2; J≦I) thereon. P j  denotes a same source-specific normally incident energy flux on each stack from source j. In each of I independent exposure steps, the I stacks are concurrently exposed to radiation from the J sources. V i  and S i  respectively denote an actual and target energy flux transmitted into the substrate via stack i in exposure step i (i=1, . . . , I). t(i) and P t(i)  are computed such that: V i  is maximal through deployment of source t(i) as compared with deployment of any other source for i=1, . . . , I; and an error E being a function of |V 1 −S 1 |, |V 2 −S 2 |, . . . , |V I −S I | is about minimized with respect to P i  (i=1, . . . , I).

RELATED APPLICATIONS

This application is related to U.S. patent application Ser. No. ______(Attorney Docket No. BUR920060017US1) entitled “SIMULTANEOUS IRRADIATIONOF A SUBSTRATE BY MULTIPLE RADIATION SOURCES”, filed on ______, andhereby incorporated by reference.

FIELD OF THE INVENTION

1. Technical Field

The present invention relates generally to irradiation of a substrateand more particularly to serial irradiation of a substrate by multipleradiation sources.

2. Background of the Invention

Rapid thermal anneal (RTA) is used in semiconductor device fabricationto heat a wafer to alter the wafer's properties, such as to activatedopants, repair damage from ion implantation, transport dopants in orout of the wafer or to other locations within the wafer, etc.

Rapid thermal anneal of a silicon wafer is often effected through directexposure of the wafer to electromagnetic radiation. Annealing is usuallyperformed after patterning of multiple stacks of dielectric layers onthe silicon wafer. When electromagnetic radiation is incident on thesestacks, constructive and destructive interference occur due toreflections at each interface in the path of the incident radiation. Asa result of the constructive and destructive interference specific toeach interface in each stack, the fraction of the incidentelectromagnetic radiation transmitted (and absorbed) into the siliconwafer is different in the vicinity of different stack-wafer interfaces.Thus the wafer regions are not heated uniformly in these circumstances.The length (L) over which thermal equilibrium is achieved can beapproximated by L˜(t*k/c_(v))^(1/2), where k and c_(v) are the thermalconductivity and specific heat of silicon, respectively, and t is thetime scale over which the incident radiation is held at a constant powerdensity. State-of-the-art thermal processing employs electromagneticradiation on time scales below 0.1 s and as a result thermal equilibriumis not achieved over length scales that are smaller than a typical VeryLarge-Scale Integration (VLSI) die size.

Thus there is a need to improve the spatial uniformity of thermalannealing of silicon wafers.

SUMMARY OF THE INVENTION

The present invention provides a method for configuring radiationsources to serially irradiate a substrate, said method comprising:

specifying J different electromagnetic sources of radiation denoted assource 1, source 2, . . . , source J, wherein each source of the Jsources is characterized by a different function of wavelength andangular distribution of its emitted radiation, said J≧2;

specifying the substrate, said substrate comprising a base layer and Istacks on the base layer, said I≧2, wherein P_(j) denotes a samenormally incident energy flux on each stack from source j such thatP_(j) is specific to source j for j=1, 2, . . . , J, wherein J≦I;

specifying a target energy flux S_(i) targeted to be transmitted viaeach stack i into the substrate such that S_(i) is specific to eachstack i for i=1, 2, . . . , I;

for serial exposure of the I stacks to radiation from the J sources suchthat the I stacks are concurrently exposed to only one source t(i) ofthe J sources in each exposure step i of I independent exposure steps,calculating t(i) and P_(t(i)) such that an actual energy flux V_(i)transmitted into the substrate via stack i in exposure step i is maximalthrough deployment of said only one source t(i) as compared withdeployment of any remaining source of the J sources for i=1, 2, . . . ,and I, and wherein an error E being a function of |V₁−S₁|, |V₂−S₂|, . .. , |V_(I)−S_(I)| is about minimized with respect to P_(i) for i=1, 2, .. . , I.

The present invention provides a method for serially irradiating asubstrate by a plurality of radiation sources, said method comprising:

providing J different electromagnetic sources of radiation, each sourceof said J sources characterized by a different function of wavelengthand angular distribution of its emitted radiation, said J≧2;

providing the substrate, said substrate comprising a base layer and Istacks on the base layer, said I≧2, wherein P_(j) denotes a samenormally incident energy flux on each stack from source j such thatP_(j) is specific to source j for j=1, 2, . . . , J;

concurrently exposing the I stacks to radiation from only one sourcet(i) of the J sources in each exposure step i of I independent exposuresteps such that either a first condition or a second condition issatisfied;

wherein the first condition is that said only one source t(i) isselected from the J sources in exposure step i such that an actualenergy flux V_(i) transmitted into the substrate via stack i in exposurestep i is maximal through deployment of said only one source t(i) ascompared with deployment of any remaining source of the J sources fori=1, 2, . . . , and I, wherein an error E being a function of |V₁−S₁|,|V₂−S₂|, . . . , |V_(I)−S_(I)| is about minimized with respect to P_(i)for i=1, 2, . . . , I, wherein S_(i) denotes a specified target energyflux targeted to be transmitted via stack i into the substrate such thatS_(i) is specific to each stack i for i=1, 2, . . . , I, wherein J≦I;

wherein the second condition is a specified design condition on thesubstrate pertaining to a device parameter of the substrate.

The present invention provides a system for serially irradiating asubstrate by a plurality of radiation sources, said substrate comprisinga base layer and I stacks on the base layer, said system comprising:

J different electromagnetic sources of radiation, each source of said Jsources characterized by a different function of wavelength and angulardistribution of its emitted radiation, said J≧2;

means for concurrently exposing the I stacks to radiation from only onesource t(i) of the J sources in each exposure step i of I independentexposure steps such that either a first condition or a second conditionis satisfied, wherein I≧2, and wherein P_(j) denotes a same normallyincident energy flux on each stack from source j such that P_(j) isspecific to source j for j=1, 2, . . . , J;

wherein the first condition is that said only one source t(i) isselected from the J sources in exposure step i such that an actualenergy flux V_(i) transmitted into the substrate via stack i in exposurestep i is maximal through deployment of said only one source t(i) ascompared with deployment of any remaining source of the J sources fori=1, 2, . . . , and I, wherein an error E being a function of |V₁−S₁|,|V₂−S₂|, . . . , |V_(I)−S_(I)| is about minimized with respect to P_(i)for i=1, 2, . . . , I, wherein S_(i) denotes a specified target energyflux targeted to be transmitted via each stack i into the substrate suchthat S_(i) is specific to each stack i for i=1, 2, . . . , I, whereinJ≦I; and

-   -   wherein the second condition is a specified design condition on        the substrate pertaining to a device parameter of the substrate.

The present invention advantageously improves the spatial uniformity ofthermal annealing of silicon wafers.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 depicts a front cross-sectional view of a substrate and radiationsources adapted to irradiate the substrate with electromagneticradiation, in accordance with embodiments of the present invention.

FIG. 2 depicts sources of electromagnetic radiation and their angulardistributions, in accordance with embodiments of the present invention.

FIG. 3 depicts the substrate of FIG. 1 with radiation from a sourceincident on a surface of the substrate in an angular distributioncharacterized by a solid angle, in accordance with embodiments of thepresent invention.

FIG. 4 illustrates that the solid angle of FIG. 3 is characterized by apolar angle and an azimuthal angle, in accordance with embodiments ofthe present invention.

FIG. 5 depicts the substrate of FIG. 1 after a dielectric film is placedon the top surface of the substrate, in accordance with embodiments ofthe present invention.

FIG. 6 is a flow chart describing a method for configuring radiationsources to serially irradiate a substrate with one radiation flux ineach of a plurality of exposure steps, in accordance with embodiments ofthe present invention.

FIG. 7 is a flow chart describing a method for serially irradiating asubstrate with one radiation flux in each of a plurality of exposuresteps, in accordance with embodiments of the present invention.

FIG. 8 depicts normally incident radiation propagating through thelayers of a stack in a substrate, in accordance with embodiments of thepresent invention.

FIG. 9 depicts radiation incident on a stack of a substrate at a solidangle, in accordance with embodiments of the present invention.

FIG. 10 depicts the substrate and radiation of FIG. 9 such that thesolid angle defines a polar angle and an azimuthal angle with respect toa rectangular coordinate system, in accordance with embodiments of thepresent invention.

FIG. 11 illustrates a computer system used for configuring radiationsources to serially irradiate a substrate, in accordance withembodiments of the present invention.

DETAILED DESCRIPTION OF THE INVENTION 1. Introduction

FIG. 1 depicts a front cross-sectional view of a substrate 10 withradiation sources 21, 22, and 23 adapted to irradiate the substrate 10with electromagnetic radiation 31, 32, and 33, respectively, inaccordance with embodiments of the present invention. The radiation 31,32, and 33 is incident on the top surface 19 of the substrate 10. Thesubstrate 10 comprises a base layer 15 and layered stacks 11, 12, and 13on and in direct mechanical contact with the base layer 15. The baselayer 15 may comprise comprises a dielectric material, a semiconductormaterial, a metal, an alloy, etc. For example, the base layer 15 may bea semiconductor layer (e.g., a semiconductor wafer) comprising asemiconductor material (e.g., single crystal silicon, polysilicon,germanium, etc.—doped or undoped). The substrate 10 may terminate withthe base layer 15. Alternatively the base layer 15 may be disposedbetween the stacks 11-13 and one or more additional layers of thesubstrate.

Stack 11 comprises a layer 11A of semiconductor material. Stack 12comprises dielectric layers 12A and 12B. Stack 13 comprises dielectriclayers 13A, 13B, and 13C. Each dielectric layer 12A, 12B, 13A, 13B, and13C independently comprises a dielectric material.

Generally, a plurality of stacks is disposed on, and in directmechanical contact with, the base layer 15. Each stack comprises one ormore layers. Each layer of each stack may independently comprise adielectric material (e.g., silicon dioxide, silicon nitride, aluminumoxide, high-k dielectric, low-k dielectric), a semiconductor material(e.g., single crystal silicon, polysilicon, germanium, etc.—doped orundoped), a metal (e.g., tungsten), an alloy (e.g., tungsten silicide),or a combination thereof. Thus, each stack has a first layer of the oneor more dielectric layers that is on and in direct mechanical contactwith the base layer 15. For example in FIG. 1, the first layers 11A,12A, and 13A of dielectric stacks 11, 12, and 13, respectively, are onand in direct mechanical contact with the base layer 15 at theinterfacial surface 14.

Each radiation source of the radiation sources 21-23 may emit radiationin any angular distribution from each source. To illustrate, FIG. 2depicts radiation sources 24-26 of electromagnetic radiation and theirangular distributions, in accordance with embodiments of the presentinvention. Source 24 emits radiation 24A in all directions. Source 25emits radiation 25A within a limited solid angular range ψ₁. Source 26emits radiation 26A unidirectionally in a direction described by a solidangle Ψ₂ with respect to a reference direction 8.

If a source emits radiation over a finite range of directions, theemitted radiation may be isotropic or anisotropic within the finiterange of directions. In addition, the sources may each be independentlymonochromatic or polychromatic with respect to the wavelength λ of theradiation. Generally, the power distribution Q(λ,Ψ) of the radiationemitted from the source, as a function of wavelength λ and solid angulardirection Ψ, may be of the form Q(λ,Ψ)=Q₀*Q₁(λ, Ψ) subject to anormalization condition of ∫∫dΨdλQ₁(λ,Ψ)=1. With the precedingnormalization, Q₀ denotes the power generated (e.g., in units ofjoule/sec) by the source. If the preceding normalization is notoperative, then Q₀ is proportional to the power generated by the source.In one embodiment, Q₁(λ,Ψ) is separable into a product of a functionQ₁₁(λ) of λ and a function Q₁₂(Ψ) of Ψ (i.e., Q₁(λ,Ψ)=Q₁₁(λ)*Q₁₂(Ψ)).For a monochromatic source having wavelength λ₀, Q₁₁(λ) may be expressedin terms of a delta function; e.g., Q₁₁(λ)˜δ(λ−λ₀). Each source ofsources 21-23 is characterized by a power distribution Q(λ,Ψ) whosegenerated power Q₀ is specific to each source and whose functionaldependence Q₁(λ,Ψ) on λ and Ψ is specific to each source. The sources21-23 differ from one another with respect to Q₁(λ,Ψ); i.e., the sourcesdiffer in the distribution of power with respect to λ, Ψ, or both λ andΨ.

Returning to FIG. 1, the electromagnetic radiation emitted by a givensource is incident upon the substrate 10 in the direction of energy flowwith an associated energy flux P. If the radiation in the direction ofenergy flow is projected onto a direction that is normal to the topsurface 19 of the surface, then the resultant energy flux P normallydirected into the stack is assumed to be stack independent (i.e., eachstack receives about the same energy flux P of radiation from a givensource). The energy flux into the stack is in units of power per unitsurface area of the top surface 19 of the stack, which is equivalent tounits of energy per unit time per unit surface area.

As explained supra, a same energy flux is incident on the differentstacks from a given source and said same energy flux on each stack isspecific to each source. However, different energy fluxes may beincident on the substrate 10 (and on the stacks 11-13) from differentsources. Similarly, a same angular distribution of radiation is incidenton the different stacks from a given source and the same angulardistribution of radiation on each stack is specific to the each source.However, different angular distributions of radiation may be incident onthe substrate 10 (and on the stacks 11-13) from different sources. Thus,the sources are geometrically distributed in relation to the stacks suchthat for each source, there is a negligible difference in the energyflux and in the angular distribution of radiation incident on eachstack.

FIG. 3 depicts the substrate 10 of FIG. 1 with radiation 29A from asource 29 incident on surface 19 of the substrate 10, in accordance withembodiments of the present invention. The source 29 may represent any ofthe sources 21-23 of FIG. 1. The source 29 emits the radiation 29A in anangular distribution in terms of the solid angular direction Ψ withrespect to the reference direction 8, and the radiation 29A is incidenton the substrate 10 in an angular distribution with respect to the solidangular direction Ω with respect to the reference direction 8. If thesource 29 emits radiation according to the power distribution Q(λ,Ψ),then the energy flux component normally incident upon the substrate 10is governed by an energy flux P (as described supra) and a distributionU(λ,Ω) in wavelength λ and solid angle Ω. Given the power distributionQ₀*Q₁(λ,Ψ) of the source as described supra, the energy flux P ofradiation incident on the substrate 10, and the distribution U(λ,Ψ) inwavelength λ and in solid angular direction Ω, may be deduced from Q₀and Q₁(λ,Ψ) by a person of ordinary skill in the art, in considerationof the location and spatial distribution of the sources 21-23 inrelation to the location and normal direction of the surface 19 of thesubstrate 10.

FIG. 4 illustrates that the solid angle Ω of FIG. 3 is characterized bya polar angle θ and an azimuthal angle Φ with respect to an XYZorthogonal coordinate system as shown, in accordance with embodiments ofthe present invention. The solid angle Ψ for the radiation emitted fromthe source 29 in FIG. 3 is similarly characterized by its polar angleand azimuthal angle (not shown).

The present invention provides a serial irradiation algorithm forconfiguring a plurality of radiation sources to irradiate a substrate byserially irradiating the substrate in a plurality of exposure steps withone and only one radiation sources in each exposure step.

2. Serial Irradiation Algorithm

The following serial irradiation algorithm provides a method forconfiguring a plurality of radiation sources to serially irradiate asubstrate in a plurality of exposure steps. The purpose of irradiatingthe substrate may be to, inter alia, anneal the base layer or a portionthereof. The serial irradiation algorithm computes the energy fluxincident on each stack of a substrate, wherein J different sources ofelectromagnetic radiation serially irradiate I stacks in each of Iexposure steps, subject to I≧2, J≧2, and J≦I. Each source of the Jsources is characterized by a different function of wavelength andangular distribution of its emitted radiation. As applied to FIG. 1, theserial irradiation algorithm has the characteristic that during eachstep of the I exposure steps, one and only one source of the Jirradiates all stacks 11-13 of the substrate 10.

For the serial irradiation algorithm, the stacks are disposed on, and indirect mechanical contact with, the base layer within the substrate.Each stack comprises one or more layers, and each layer of each stackmay independently comprise a dielectric material, a semiconductormaterial, a metal, an alloy, or a combination thereof. The substrate mayterminate with the base layer. Alternatively the base layer may bedisposed between the stacks and one or more additional layers.

P_(j) denotes the component of the energy flux that is normally incidenton each stack i and originates from source j (j=1, 2, . . . , J) inexposure step i (i=1, 2, . . . , I). The choice of j for the one andonly source selected in each exposure step i (i=1, 2, . . . , I) isdenoted as t(i) and satisfies a selection criterion to be indicatedinfra. Generally, the source t(i) is independently chosen for eachexposure step i. Thus t(i) is a function of i.

S_(i) denotes a target energy flux transmitted in exposure step i intothe substrate 10 via stack i (i=1, 2, . . . , I) from source t(i). Afirst portion of S_(i) may be absorbed within stack i and a secondportion of S_(i) may be transmitted through stack i to enter the baselayer 15. In one embodiment, the first portion of S_(i) is negligible incomparison with the second portion of S_(i). S_(i) is an input to theserial irradiation algorithm. In one embodiment, a goal is to focus onhow the target energy fluxes S_(i) (i=1, 2, . . . , I) are related toone another rather than on their magnitudes individually. Therefore,S_(i) may be provided as input in a normalized form such being subjectto Σ_(i)S_(i)=1, wherein the summation Σ_(i) is from i=1 to i=I.

T_(ij) is a transmission coefficient for stack i relative to energy fluxP_(j). In particular, T_(ij) is the fraction of energy flux P_(j) thatis transmitted via stack i into the substrate 10. T_(ij) may bedetermined experimentally, or by calculation as described infra.

In exposure step i (i=1, 2, . . . , I), one and only one source, namelysource t(i), of the J sources is selected to concurrently irradiate theI stacks. Source t(i) satisfies the constraint that the choice of t(i)to determine P_(j), as compared with a choice of any other source of theJ sources, maximizes T_(ij)P_(j) (j=1, 2, . . . , J) in each exposurestep i. Let V_(i) be defined such that

V _(i) =T _(i,t(i)) P _(t(i))  (1)

-   -   wherein V_(i) represents the maximum value of T_(ij)P_(j) in        exposure step i through choice of source j=t(i) of the J        sources.

Let W_(u) denote the actual energy flux transmitted into the substrate10 via stack u in exposure step i during which the I stacks are exposedto source t(i), wherein u=1, 2, . . . , I. A first portion of W_(u) maybe absorbed within stack u and a second portion of W_(u) may betransmitted through stack u to enter the base layer 15. In oneembodiment, the first portion of W_(u) is negligible in comparison withthe second portion of W_(u). Thus,

W _(u) =T _(u,t(i)) P _(t(i)) (u=1, 2, . . . , I)  (2)

-   -   Note that W_(i)=V_(i).

Given S_(i) (e.g., via user input), the serial irradiation algorithmdetermines the energy fluxes P_(t(i)) in exposure step i (i=1, 2, . . ., I) so as to closely match V_(i) to S_(i). In particular the serialirradiation algorithm minimizes an error E which is a function of|V₁−S₁|, |V₂−S₂|, . . . , |V_(I)−S_(I)|. For example, E may have thefunctional form:

E=Σ _(i) |V _(i) −S _(i)|^(B)  (3A)

wherein B is a positive real number, and wherein the summation Σ_(i) isfrom i=1 to i=I. In one embodiment B=2, resulting in E being expressedas:

E=Σ _(i)(V _(i)−S_(i))²  (3B)

For illustrative purposes, the following discussion will employ Equation(3B) for E. However, as stated supra, the scope of the present inventiongenerally considers E to be a function of |V₁−S₁|, |V₂−S₂|, . . . ,|V_(I)−S_(I)| such as in the embodiment of Equation (3A). SubstitutingV_(i) of Equation (1) into Equation (3B):

E=Σ _(i)(T _(i,t(i)) P _(t(i)) −S _(i))  (4)

Thus given T_(ij) and S_(i) (i=1, 2, . . . , I and j=1, 2, . . . , J)which are inputs to the serial irradiation algorithm, a minimization ofE in Equation (4) determines both t(i) and P_(t(i)) for i=1, 2, . . . ,I. The preceding minimization of E of Equation (4) is defined asminimizing E with respect to arbitrarily small variations in P_(t(i))subject to the constraint of determining j=t(i) so as to maximizeT_(ij)P_(j) (j=1, 2, . . . , J) in each exposure step i (i=1, 2, . . . ,I). The preceding mathematical problem may be solved by any method knownto a person of ordinary skill in the art. For example, the known simplexmethod may be employed. References for the simplex method comprise: R.Shamir, “The efficiency of the simplex method: a survey” ManagementScience, 33 :3 (1987) pp. 301-334; andhttp://en.wikipedia.org/wiki/Simplex_algorithm.

In the solution for t(i), i=1, 2, . . . , I, it is possible for the samet(i) to represent the same source in two or more exposure steps (i.e.,it is possible for t(i1)=t(i2) for i1≠i2). Hence, J≦I generally.

After the energy source t(i) and the associated energy fluxes P_(t(i))for i=1, 2, . . . , I are computed, the computed values of t(i) andP_(t(i)) for i=1, 2, . . . , I may be substituted into Equation (4) tocompute the error E. In some embodiments, E=0 (i.e., V_(i)=S_(i) for i=,1, . . . , I).

However, the error E may be non-zero and could be compared with amaximum acceptable error E_(MAX) for acceptability. If E>E_(MAX), thenseveral remedies may be available. These remedies change the model ofthe radiation sources. The currently used sources could be changed withrespect to their characteristics and execution of the preceding serialirradiation algorithm is then repeated (e.g., by change of powerspectrum and/or angular distribution of radiated power for polychromaticsources; by change of wavelength and/or the direction of energypropagation for monochromatic sources). Alternatively, one or moreadditional sources could be added. Another source model change is toboth change source characteristics of the currently used sources and addone or more additional sources.

As another remedy, as depicted in FIG. 5 in accordance with embodimentsof the present invention, a dielectric film 16 could be placed on thetop surface 19 of the substrate 10 (either on the entire surface 19 asshown or on top of selected stacks i) in order to increase thetransmission coefficient T_(ij) of those stacks i through which theincident energy flux P_(j) is otherwise poorly transmitted. Thedielectric film 16 comprises layers 16A, 16B, and 16C, which serve asextensions of stacks 11, 12, and 13, respectively. Thus, thetransmission coefficient T_(ij) of each stack i is changed by theaddition of the layers 16A, 16B, and 16C to the stacks 11, 12, and 13,respectively. For example, if layer 11A of stack 11 comprisessemiconductor material which is characterized by poor transmission ofelectromagnetic radiation through the stack 11, then the addition oflayer 16A to stack 11 may substantially increase the transmissioncoefficient of stack 11. With the addition of the dielectric film 16,surface 19A replaces surface 19 as the top surface of the substrate 10.

An attempt to solve the preceding mathematical problem for t(i) andP_(t(i)) for i=1, 2, . . . , I could have other associated problems. Forexample, there may be no unique solution for the preceding mathematicalproblem. As another example, the solution for P_(j) may be non-physical;i.e., at least one of the energy fluxes P_(t(i)) is computed to benegative. Generally, any of the preceding problematic cases (i.e.,unacceptable error, no solution, non-physical solution) may trigger anyof the preceding remedies followed by repetition of execution of thepreceding serial irradiation algorithm.

After t(i) and P_(t(i)) (i=1, 2, . . . , I) are successfully calculatedas described supra, the powers Q_(0,t(i)) (i=1, 2, . . . , I) at thesources t(i) (i=1, 2, . . . , I) corresponding to exposure step i may becomputed from P_(t(i)) (i=1, 2, . . . , I) and geometric relationshipsbetween these sources and the top surface 19 of the substrate 10 asdescribed supra.

The I exposure steps may be serially performed in any time sequence withrespect to the exposure sources. As an example with three stacks, thethree stacks may be concurrently exposed at a first time TIME(1) tosource t(2), at a second time TIME(2) to source t(1), and at a thirdtime TIME(3) to source t(3), wherein TIME(1)<TIME(2)<TIME(3). As anotherexample with three stacks, the three stacks may be concurrently exposedat a first time TIME(1) to source t(1), at a second time TIME(2) tosource t(2), and at a third time TIME(3) to source t(3). The precedingserial irradiation algorithm is summarized in the flow chart of FIG. 6.

FIG. 6 is a flow chart describing a method for configuring radiation toserially irradiate a substrate with one radiation flux in each of aplurality of exposure steps, in accordance with embodiments of thepresent invention. The method of FIG. 6 computes the energy sources t(i)and associated energy fluxes P_(t(i)) (i=1, 2, . . . , I) incident onthe I stack of a substrate in I serial exposure steps, wherein J sourcesof electromagnetic radiation serially irradiate I stacks, subject toI≧2, J≧2, and J≦I.

Step 61 provides input to the serial irradiation algorithm, said inputcomprising: J electromagnetic sources of radiation (J≧2); a substrate;target energy fluxes S_(i) (i=1, 2, . . . , I); and transmissioncoefficients T_(ij). Each source of the J sources is characterized by adifferent function of wavelength and angular distribution of its emittedradiation. The input for the J electromagnetic sources includesspecification of the distribution Q_(j)(λ,Ψ) in wavelength λ and solidangle Ψ of the emitted radiation for each source j (j=1, 2, . . . , J).

The substrate comprises a base layer and I stacks thereon. Each stackcomprises at least one layer such that a first layer of the at least onelayer is on and in direct mechanical contact with the base layer. Eachlayer of each stack may independently comprise a dielectric material, asemiconductor material, a metal, an alloy, or a combination thereof.P_(j) denotes a same normally incident energy flux on each stack fromsource j such that P_(j) is specific to source j for j=1, 2, . . . , J.W_(i) denotes an actual energy flux transmitted into the substrate viastack i in exposure step 1 for i=1, 2, . . . , and I. The target energyflux S_(i) is targeted to be transmitted via of each stack i into thesubstrate in exposure step i such that S_(i) is specific to each stack ifor i=1, 2, . . . , I. The transmission coefficients T_(ij) are definedas the fraction of the energy flux P_(j) that is transmitted via stack iinto the substrate. W_(u) is the energy flux transmitted into thesubstrate via stack u in irradiation step i, whereinW_(u)=T_(u,t(i))P_(t(i)) (u=1, 2, . . . , I). Note that W_(i)=V_(i).

For serial exposure of the I stacks to radiation from the J sources suchthat the I stacks are concurrently exposed to only one source t(i) ofthe J sources in each exposure step i of I independent exposure steps,t(i) and P_(t(i)) are computed such that an actual energy flux V_(i)transmitted into the substrate via stack i in exposure step i is maximalthrough deployment of said only one source t(i) as compared withdeployment of any remaining source of the J sources for i=1, 2, . . . ,and I, wherein V_(i)=T_(i,t(i))P_(t(i)), and wherein a summation over ifrom i=1 to i=I of (V_(i)−S_(i))² is about minimized with respect toarbitrarily small variations in P_(j(i)) for i=1, 2, . . . , I.

The calculation of t(i) and P_(t(i)) (i=1, 2, . . . , I) in step 62 maybe successful or may not be successful for the reasons stated supra(i.e., unacceptable error, no solution, non-physical solution). Step 63determines whether the calculation of t(i) and P_(t(i)) (i=1, 2, . . . ,I) in step 62 was successful

If step 63 determines that the calculation of t(i) and P_(t(i)) (i=1, 2,. . . , I) in step 62 was not successful, then step 64 changes the modelin any manner that has been described supra (i.e., changing sourcecharacteristics of one or more sources and/or adding one or moreadditional sources and/or placing a dielectric film on the top surfaceof the substrate), followed by iteratively looping back to step 61 torepeat performance of steps 61-64 until step 63 determines that thecalculation of t(i) and P_(t(i)) (i=1, 2, . . . , I) in step 62 wassuccessful or until a maximum specified number of iterations of steps61-64 has been performed. As explained supra, changing sourcecharacteristics of one or more sources may be implemented by change ofpower spectrum and/or angular distribution of radiated power which maybe implemented: for polychromatic sources; by changing at least onepolychromatic source to a monochromatic source; by changing at least onemonochromatic source to a polychromatic source; by changing wavelengthand/or the direction of energy propagation for monochromatic sources;etc. Note that the only input in step 61 that would need to be providedafter executing step 64 is the input that has been changed for thecurrent iteration (e.g., input related to model changes in the precedingexecution of step 64).

If step 63 determines that the calculation of t(i) and P_(t(i)) (i=1, 2,. . . , I) in step 62 was successful, then step 65 calculates the sourcepower Q_(0,t(i)) (i=1, 2, . . . , I) at the sources t(i) (i=1, 2, . . ., I) corresponding to exposure step i from the calculated P_(t(i)) (i=1,2, . . . , I). As discussed supra, the source power may be deduced fromthe calculated P_(t(i)) and the inputted Q_(t(i))(λ,Ψ) (i=1, 2, . . . ,I).

The method of FIG. 6 could end with performance of step 65.Alternatively, step 66 may be performed after step 65 is performed.

In step 66, the computed source powers Q_(0,t(i)) (i=1, 2, . . . , I)are adjusted or tuned to satisfy a design condition on the substrate 10.For example, the adjusted source powers may be obtained by experimentalevaluation, comprising performing anneal experiments on wafers andmeasuring device parameters such as transistor threshold voltage,extrinsic resistance, or drive current, or other device parameters suchas doped-silicon sheet resistances. The source powers Q_(0j) may bevaried individually or in aggregate to reflect desired parametersmeasured at multiple locations having varying average stackcompositions, wherein the adjusted source powers enable achievement ofspecified uniformity or conformance to specified variation in thepreceding device parameters.

FIG. 7 is a flow chart describing a method for serially irradiating asubstrate with one radiation flux in each of a plurality of exposuresteps, in accordance with embodiments of the present invention. Themethod of FIG. 7 utilizes the source powers computed in accordance withthe preceding serial irradiation algorithm of FIG. 6.

Step 71 provides J electromagnetic sources of radiation, wherein J≧2.Each source of the J sources is characterized by a different function ofwavelength and angular distribution of its emitted radiation.

Step 72 provides a substrate. The substrate comprises a base layer and Istacks thereon, wherein I≧2 and J≦I. Each stack comprises at least onelayer such that a first layer of the at least one layer is on and indirect mechanical contact with the base layer. Each layer of each stackmay independently comprise a dielectric material, a semiconductormaterial, a metal, an alloy, or a combination thereof. P_(j) denotes asame normally incident energy flux on each stack from source j such thatP_(j) is specific to source j for j=1, 2, . . . , J.

Step 73 concurrently exposes the I stacks to radiation from only onesource t(i) of the J sources in each exposure step i of I independentexposure steps such that either a first condition or a second conditionis satisfied.

The first condition is that said only one source t(i) is selected fromthe J sources in exposure step i such that an actual energy flux V_(i)transmitted into the substrate via stack i in exposure step i is maximalthrough deployment of said only one source t(i) as compared withdeployment of any remaining source of the J sources for i=1, 2, . . . ,and I, wherein a summation over i from i=1 to i=I of (V_(i)−S_(i))² isabout minimized with respect to arbitrarily small variations in P_(t(i))for i=1, 2, . . . , I. S_(i) denotes a specified target energy fluxtargeted to be transmitted via stack i into the substrate such thatS_(i) is specific to each stack i for i=1, 2, . . . , I, wherein J≦I.V_(i)=T_(i,t(i))P_(t(i)), and T_(i,t(i)) is defined as the fraction ofthe energy flux P_(t(i)) that is transmitted via stack i into thesubstrate for i=1, 2, . . . , I.

The second condition is a specified design condition on the substratepertaining to a device parameter of the substrate such as transistorthreshold voltage, extrinsic resistance, or drive current, or otherdevice parameters such as doped-silicon sheet resistances as describedsupra in conjunction with step 66 of FIG. 6.

To illustrate the result of executing the serial irradiation algorithmas applied to FIG. 1 for irradiating the substrate 10, assume thatsources 22, 21, and 23 are utilized in steps 1, 2, and 3, respectively.In step 1, source 22 and only source 22 is turned on to irradiate thesubstrate 10. In step 2, source 21 and only source 21 is turned on toirradiate the substrate 10. In step 3, source 23 and only source 23 isturned on to irradiate the substrate 10.

As an example, consider the following transmission matrix T_(ij) inTable 1 for application of the serial irradiation algorithm, using threesources (Source 1, Source 2, Source 3) and three stacks (Stack 1, Stack2, Stack 3).

TABLE 1 T_(ij) Source 1 Source 2 Source 3 Stack 1 0.6 0.2 0.3 Stack 20.5 0.5 0.4 Stack 3 0.3 0.04 0.6

Given the target energy fluxes of S₁=S₂=S₃=1.0, the solution for P_(j)in this example is: P₁=1.666667, S₂=2.0, S₃=1.666667, as may beconfirmed in Table 2 by inspection.

TABLE 2 TijPj Source 1 Source 2 Source 3 Max (TijPj) Stack 1 1.0 0.4 0.51.0 Stack 2 0.8333 1.0 0.666664 1.0 Stack 3 0.5 0.08 1.0 1.0

The serial irradiation algorithm may be employed in an embodiment inwhich a pertinent characteristic of a region within the substrate 10 iscontrolled by the highest temperature that the region is exposed to fromradiation emitted by the radiation sources 21-23 during an annealingprocess, irrespective of a prior and/or subsequent lower temperaturethat the region within the substrate 10 is exposed to.

3. Determination of Transmission Coefficients

Let the energy flux incident on the substrate from a given radiationsource be described as a distribution U(λ,Ω) in wavelength λ and solidangle Ω. Let T(λ,Ω) denote the transmission coefficient of a stack for agiven wavelength λ and solid angle Ω. T(λ,Ω) may be determinedexperimentally by experimental techniques known to a person of ordinaryskill in the art. Alternatively, T(λ,Ω) may be calculated for eachspecified combination of λ and Ω, as will be described infra. AfterT(λ,Ω) is determined, either experimentally or by calculation, theintegrated transmission coefficient T for the stack may be calculatedvia

T=∫∫dΩdλU(λ,Ω)T(λ,Ω)/∫∫dΩdλU(λ,Ω)  (5)

The integrations in Equation (5) are performed over the range ofwavelength λ and solid angle Ω for which U(λ,Ω) is defined. The role ofU(λ,Ω) in Equation (5) is that of a weighting function reflecting howthe incident radiation from the given source is distributed in bothwavelength λ and solid angle Ω.

Alternatively, T for each stack receiving radiation from the givenradiation source may be determined experimentally by experimentaltechniques known to a person of ordinary skill in the art.

T(λ,Ω) may be computed for a given combination of λ and Ω, wherein Ω esuch that the radiation incident upon the substrate 10 may be normal ornon-normal to the substrate 10. In Section 3.1, an algorithm forcalculating T(λ,Ω) will be described under the assumption that theradiation is normally incident upon the substrate. In Section 3.2, analgorithm for calculating T(λ,Ω) will be described for any solid angleof incidence Ω.

3.1 Normal Incidence of Radiation

T(λ,Ω) may be computed for a given combination of λ and Ω under theassumption that the radiation is normally incident upon the substrate.For normal incidence of the radiation at a specified wavelength λ, FIG.8 depicts the incident radiation 79 propagating through the layers of astack in the substrate 10, in accordance with embodiments of the presentinvention. In FIG. 8, the stack comprises N−1 layers denoted as layers1, 2, . . . , N−1. Layer 0 characterizes the medium from which theradiation 79 enters the stack at top surface 19. Layer N represents thebase layer 15 into which the radiation is transmitted from layer N−1 ofthe stack at interfacial layer 14. In FIG. 8, n_(m) denotes the index ofrefraction of layer m, z_(m) denotes the coordinate value in thedirection 17 at the interface between layer m and layer m+1 (m=0, 1, . .. , N−1) subject to z₀=0. F_(m) denotes the forward electric fieldcomplex amplitude in layer m for the radiation propagating in thedirection 17. R_(m) denotes the reflected electric field complexamplitude in layer m for the reflected radiation propagating in thedirection 18 which is opposite the direction 17 (m=0, 1, . . . , N).Physically, the reflected components R_(m) are generated by thediscontinuity in index of refraction at the interfaces (i.e., betweenlayers m−1 and m for m=1, 2, . . . , N).

Continuity of the electric field and its derivative at the interfacebetween layers m−1 and m (m=1, 2, . . . , N) respectively results in thefollowing equations:

F _(m-1)exp(ik _(m-1) z _(m-1))+R _(m-1)exp(−ik _(m-1) z _(m-1))=F_(m)exp(ik _(m) z _(m))+R _(m)exp(−ik _(m) −z _(m))  (6)

k _(m-1) F _(m-1)exp(ik _(m-1) z _(m-1))−k _(m-1) R _(m-1)exp(−ik _(m-1)z _(m-1))=k _(m) F _(m)exp(ik _(m) z _(m))−k _(m) R _(m)exp(−ik _(m) −z_(m))  (7)

where k_(m)=1/(2πn_(m)λ). Note that “i” in exp(±ik_(m)z_(m)) denote thesquare root of −1 and should not be confused with the use of “i” as asubscript in the description of the present invention herein.

Exemplary boundary conditions are F₀=1 and R_(N)=0. For the precedingexemplary boundary conditions, Equations (6)-(7) provide 2N linearequations and there are 2N unknowns (F₁, . . . , F_(N), R₀, . . . ,R_(N-1)) which may be solved by any method known to a person of ordinaryskill in the art (e.g., matrix inversion).

The resultant transmission coefficient T is calculated asT=(1−|R₀|²)/|F₀|²; i.e. or T=1−|R₀|² for the assumed F₀=1 with thepreceding exemplary boundary conditions. Note that the assumption ofF₀=1 is arbitrary and any numerical value could have been chosen for F₀,since the transmission coefficient is the fraction of transmitted energyflux and therefore does not depend on the magnitude of F₀.

The preceding exemplary boundary condition of R_(N)=0 may occur if allof the radiation entering layer N through the stack shown in FIG. 8 isabsorbed in layer N. Alternative embodiments may be characterized byR_(N)≠0, which can be treated in a similar manner as with the R_(N)=0embodiment described supra, by setting the reflection coefficient tozero in a layer N′>N in which no reflections occur and adding additionalequations, similar to Equations (6)-(7), for layers N+1, . . . , N′.Layer N′ represents the medium (e.g., air) just below and in directmechanical contact with the substrate, as occurs in at least twoadditional embodiments. In the first additional embodiment, layer N is aterminating layer of the substrate (i.e., a bottom layer of thesubstrate), so that N′=N+1. In the second additional embodiment, thesubstrate comprises additional layers below layer N, so that N′>N+1.

3.2 Angular Incidence of Radiation

FIG. 9 depicts radiation 80 as incident on a stack of the substrate 10at a solid angle Ω with respect to a direction 17, in accordance withembodiments of the present invention. In FIG. 9, the stack comprises N−1layers denoted as layers 1, 2, . . . , N−1. Layer 0 characterizes themedium from which the radiation enters the stack at top surface 19, andlayer N represents the base layer 15 into which the radiation istransmitted from layer N−1 of the stack at interfacial layer 14.

FIG. 10 depicts the substrate 10 and radiation 80 of FIG. 9 such thatthe solid angle Ω defines a polar angle θ and an azimuthal angle Φ withrespect to a rectangular coordinate system having orthogonal axes X, Y,and Z, in accordance with embodiments of the present invention. Theplane 81 is normal to the incident radiation 80. The electric andmagnetic field vectors associated with the radiation 80 are in the plane81. The plane 81 intersects the substrate 10 in the line 82 which is theX axis of the coordinate system. The Y axis is in the plane of the topsurface 19 of the substrate 10 and is orthogonal to the X axis. The Zaxis is orthogonal to the plane of the top surface 19 of the substrate10. Thus the polar angle θ is the angle between the direction of theradiation 80 and the Z axis. The X axis is in both the plane of thesubstrate 10 and the plane 81. The Y axis is in the plane of thesubstrate 10 and at an angle θ with respect to the plane 81.

Let F be a vector representing the forward electric field (in the plane81) associated with the incident radiation 80 into the substrate 10,wherein F denotes the magnitude of F. Let F_(Z) be a vector representingthe electric field projected onto the Z axis, wherein F_(Z) denotes themagnitude of F_(Z) . Let F_(S) be a vector representing the electricfield projected onto the plane of the top surface 19 of the substrate10. The azimuthal angle Φ is the angle between the X axis and F_(S) asshown. Let F_(X) and F_(Y) denote the magnitude of the projection ofF_(S) onto the X axis and the Y axis, respectively. Based on thepreceding definitions, F_(X), F_(Y), and F_(Z) are related to F, θ, andΦ via:

F_(X)=F cos θ cos Φ  (8)

F_(Y)=F cos θ sin Φ  (9)

F_(Z)=F sin θ  (10)

In FIGS. 9-10, the forward component and reverse component of theradiation 80 are associated with the directions 17 and 18, respectively.The index of refraction of layer m is n_(m), and z_(m) denotes thecoordinate value along the Z axis in the direction 17 at the interfacebetween layer m and layer m+1 (m=0, 1, . . . , N−1), wherein z₀=0.F_(X,m), F_(Y,m), and F_(Z,m) denote the electric field complexamplitude in the X, Y, and Z direction, respectively, in layer m (m=0,1, . . . , N) for the forward component of the radiation 80. R_(X,m),R_(Y,m), and R_(Z,m) denote the electric field complex amplitude in theX, Y, and Z direction, respectively, in layer m (m=0, 1, . . . , N) forthe reverse component of the radiation 80. Physically, the reflectedcomponents R_(X,m), R_(Y,m), and R_(Z,m) are generated by thediscontinuity in index of refraction at the interfaces (i.e., betweenlayers m−1 and m for m=1, 2, . . . , N). Note that in the descriptioninfra, the upper case symbols X, Y, Z denote coordinates axes, whereasthe lower case symbols x, y, z denote the coordinate values respectivelycorresponding to the coordinates axes X, Y, Z.

Continuity of the X component of electric field and its derivative atthe interface between layers m−1 and m (m=1, 2, . . . , N) respectivelyresults in the following equations:

F _(X,m-1)exp(ik _(m-1) z _(m-1))+R _(X,m-1)exp(−ik _(m-1) z _(m-1))=F_(X,m)exp(ik _(m) z _(m))+R _(X,m)exp(−ik _(m) −z _(m))  (11)

k _(m-1) F _(X,m-1)exp(ik _(m-1) z _(m-1))−k _(m-1) R _(X,m-1)exp(−ik_(m-1) z _(m-1))=k _(m) F _(X,m)exp(ik _(m) z _(m))−k _(m) R_(X,m)exp(−ik _(m) z _(m))  (12)

wherein k_(m)=1/(2πn_(m)λ).

Continuity of the Y component of electric field and its derivative atthe interface between layers m−1 and m (m=1, 2, . . . , N) respectivelyresults in the following equations:

F _(Y,m-1)exp(ik _(m-1) z _(m-1))+R _(Y,m-1)exp(−ik _(m-1) z _(m-1))=F_(Y,m)exp(ik _(m) z _(m))+R _(Y,m)exp(−ik _(m) z _(m))  (13)

k _(m-1) F _(Y,m-1)exp(ik _(m-1) z _(m-1))−k _(m-1) R _(Y,m-1)exp(−ik_(m-1) z _(m-1))=k _(m) F _(Y,m)exp(ik _(m) z _(m))−k _(m) R_(Y,m)exp(−ik _(m) z _(m))  (14)

Let D_(Z,m) denote the displacement complex amplitude in the Z directionin layer m and let ∈_(m) denote the permittivity of layer m (m=0, 1, . .. , N). D_(Z.m)=n² _(m)F_(Z,m), since D_(Z.m)=∈_(m)F_(Z,m) andn_(m)=(e_(m))^(1/2) (m=0, 1, 2, . . . , N). Therefore, continuity of theZ component of the displacement and its derivative in the direction Z(i.e., direction 17) at the interface between layers m−1 and m (m=1, 2,. . . , N) respectively results in the following equations:

n ² _(m-1)(F _(Z,m-1)exp(ik _(m-1) z _(m-1))+R _(Z,m-1)exp(−ik _(m-1) z_(m-1)))=n ² _(m)(F _(Z,m)exp(ik _(m) z _(m))+R _(Z,m)exp(−ik _(m) −z_(m)))  (15)

n ² _(m-1)(k _(m-1) F _(Z,m-1)exp(ik _(m-1) z _(m-1))−k _(m-1) R_(Z,m-1)exp(−ik _(m-1) z _(m-1)))=n ² _(m)(k _(m) F _(Z,m)exp(ik _(m) z_(m))−k _(m) R _(Z,m)exp(−ik _(m) z _(m)))  (16)

Equations (8)-(10) provide boundary conditions of F_(X,0)=F₀ cos θ cosΦ, F_(Y,0)=F₀ cos θ sin Φ, and F_(Z,0)=F₀ sin θ, for a given electricfield magnitude F₀ in layer 0. Additional boundary conditions which maybe employed are R_(X,N)=R_(Y,N)=R_(Z,N)=0. For the preceding boundaryconditions, Equations (11)-(16) provide 6N linear equations and thereare 6N unknowns (F_(X,1), F_(Y,1), F_(Z,1), . . . , F_(X,N), F_(Y,N),F_(Z,N), R_(X,0), R_(Y,0), R_(Z,0), . . . , R_(X,N-1), R_(Y,N-1),R_(Z,N-1)) which may be solved by any method (e.g., matrix inversion)known to a person of ordinary skill in the art.

The resultant transmission coefficient T is calculated asT=(1−|R_(X,0)|²−|R_(Y,0)|²−|R_(Z,0)|²)/(|F_(X,0)|²+(|F_(Y,0)|²+(|F_(Z,0)|²).However, |F_(X,0)|²+(|F_(Y,0)|²+(|F_(Z,0)|²=|F₀|² from the precedingboundary conditions of F_(X0)=F cos θ cos Φ, F_(Y0)=F cos θ sin Φ, andF_(Z0)=F sin θ. Therefore T=(1−|R_(X,0)|²−|R_(Y,0)|²−|R_(Z,0)|²)/|F₀|².Note that the value F₀ (e.g., F₀=1) is arbitrary and any numerical valuecould have been chosen for F₀, since the transmission coefficient is thefraction of transmitted energy flux and therefore does not depend on themagnitude of F₀.

The preceding exemplary boundary condition of R_(X,N)=R_(Y,N)=R_(Z,N)=0may occur if all of the radiation entering layer N through the stackshown in FIG. 9 is absorbed in layer N. Alternative embodiments may becharacterized by a non-zero value in at least one of R_(X,N), R_(Y,N),and R_(Z,N). These alternative embodiments can be treated in a similarmanner as with the R_(X,N)=R_(Y,N)=R_(Z,N)=0 embodiment described supra,by setting the reflection coefficients R_(X,N′), R_(Y,N′), and R_(Z,N′)to zero in a layer N′>N in which no reflections occur and addingadditional equations, similar to Equations (11) and (16), for layersN+1, . . . , N′. Layer N′ represents the medium (e.g., air) just belowand in direct mechanical contact with the substrate, as occurs in atleast two additional embodiments. In the first additional embodiment,layer N is a terminating layer of the substrate (i.e., a bottom layer ofthe substrate), so that N′=N+1. In the second additional embodiment, thesubstrate comprises additional layers below layer N, so that N′>N+1.

4. Computer System

FIG. 11 illustrates a computer system 90 used for configuring radiationsources to serially irradiate a substrate, in accordance withembodiments of the present invention. The computer system 90 comprises aprocessor 91, an input device 92 coupled to the processor 91, an outputdevice 93 coupled to the processor 91, and memory devices 94 and 95 eachcoupled to the processor 91. The input device 92 may be, inter alia, akeyboard, a mouse, etc. The output device 93 may be, inter alia, aprinter, a plotter, a computer screen, a magnetic tape, a removable harddisk, a floppy disk, etc. The memory devices 94 and 95 may be, interalia, a hard disk, a floppy disk, a magnetic tape, an optical storagesuch as a compact disc (CD) or a digital video disc (DVD), a dynamicrandom access memory (DRAM), a read-only memory (ROM), etc. The memorydevice 95 includes a computer code 97 which is a computer program thatcomprises computer-executable instructions. The computer code 97includes an algorithm for configuring radiation sources to seriallyirradiate a substrate with multiple radiation sources as describedsupra. The processor 91 executes the computer code 97. The memory device94 includes input data 96. The input data 96 includes input required bythe computer code 97. The output device 93 displays output from thecomputer code 97. Either or both memory devices 94 and 95 (or one ormore additional memory devices not shown in FIG. 11) may be used as acomputer usable medium (or a computer readable medium or a programstorage device) having a computer readable program embodied thereinand/or having other data stored therein, wherein the computer readableprogram comprises the computer code 97. Generally, a computer programproduct (or, alternatively, an article of manufacture) of the computersystem 90 may comprise said computer usable medium (or said programstorage device).

While FIG. 11 shows the computer system 90 as a particular configurationof hardware and software, any configuration of hardware and software, aswould be known to a person of ordinary skill in the art, may be utilizedfor the purposes stated supra in conjunction with the particularcomputer system 90 of FIG. 12. For example, the memory devices 94 and 95may be portions of a single memory device rather than separate memorydevices.

While embodiments of the present invention have been described hereinfor purposes of illustration, many modifications and changes will becomeapparent to those skilled in the art. Accordingly, the appended claimsare intended to encompass all such modifications and changes as fallwithin the true spirit and scope of this invention.

1. A method for configuring radiation sources to serially irradiate asubstrate, said method comprising: specifying J differentelectromagnetic sources of radiation denoted as source 1, source 2, . .. , source J, wherein each source of the J sources is characterized by adifferent function of wavelength and angular distribution of its emittedradiation, said J≧2; specifying the substrate, said substrate comprisinga base layer and I stacks on the base layer, said I≧2, wherein P_(j)denotes a same normally incident energy flux on each stack from source jsuch that P_(j) is specific to source j for j=1, 2, . . . , J, whereinJ≦I; specifying a target energy flux S_(i) targeted to be transmittedvia each stack i into the substrate such that S_(i) is specific to eachstack i for i=1, 2, . . . , I; for serial exposure of the I stacks toradiation from the J sources such that the I stacks are concurrentlyexposed to only one source t(i) of the J sources in each exposure step iof I independent exposure steps, calculating t(i) and P_(t(i)) such thatan actual energy flux V_(i) transmitted into the substrate via stack iin exposure step i is maximal through deployment of said only one sourcet(i) as compared with deployment of any remaining source of the Jsources for i=1, 2, . . . , and I, and wherein an error E being afunction of |V₁−S₁|, |V₂−S₂|, |V_(I)−S_(I)| is about minimized withrespect to P_(i) for i=1, 2, . . . , I.
 2. The method of claim 1,wherein E=Σ_(i)(V_(i)−S_(i))², wherein Σ_(i) denotes a summation over ifrom i=1 to i=I.
 3. The method of claim 1, wherein the method furthercomprises specifying transmission coefficients T_(ij), wherein T_(ij) isthe fraction of the energy flux P_(j) that is transmitted via stack iinto the substrate, and wherein V_(i)=T_(i,t(i))P_(t(i)), for i=1, 2, .. . , I.
 4. The method of claim 1, wherein the method further comprisesanalyzing said error, said analyzing comprising: ascertaining whether Eexceeds a specified maximum error E_(MAX); if said ascertainingascertains that E does not exceeds E_(MAX) then ending said method; ifsaid ascertaining ascertains that E exceeds E_(MAX) then modifying thesubstrate or modifying the sources, followed by iteratively performingsaid computing and said analyzing until E does not exceeds E_(MAX) oruntil a maximum specified number of iterations of said computing andsaid analyzing has been performed, wherein said modifying the substratecomprises adding a dielectric layer to a top surface of the substratesuch that the added dielectric layer is directly exposed to theradiation, and wherein said modifying the sources comprises replacingsaid J sources with J′ sources such that J′≧2 and said J′ sourcescollectively differ from said J sources.
 5. The method of claim 4,wherein said ascertaining ascertains that E does not exceeds E_(MAX)during a unique iteration of said computing and said analyzing, whereinsaid modifying the sources is performed during the unique iteration, andwherein said replacing the J sources with J′ sources during the uniqueiteration is subject to J′>J.
 6. The method of claim 4, wherein saidascertaining ascertains that E does not exceeds E_(MAX) during a uniqueiteration of said computing and said analyzing, wherein said modifyingthe sources is performed during the unique iteration, and wherein saidreplacing the J sources with J′ sources during the unique iteration issubject to J′=J such that the power spectrum with respect to wavelengthof at least one source of the J sources is changed during the uniqueiteration.
 7. The method of claim 4, wherein said ascertainingascertains that E does not exceeds E_(MAX) during a unique iteration ofsaid computing and said analyzing, wherein said modifying the sources isperformed during the unique iteration, and wherein said replacing the Jsources with J′ sources during the unique iteration is subject to J′=Jsuch that the angular distribution of radiated power of at least onesource of the J sources is changed during the unique iteration.
 8. Themethod of claim 1, wherein the method further comprises computing thesource power of each source of the J sources from the computed energyfluxes P_(j) (j=1, 2, . . . , J).
 9. The method of claim 8, wherein themethod further comprises adjusting the computed source power of the Jsources to satisfy a specified design condition on the substrate.
 10. Acomputer program product, comprising a computer usable medium having acomputer readable program code embodied therein, said computer readableprogram code comprising an algorithm adapted to implement the method ofclaim
 1. 11. A method for serially irradiating a substrate by aplurality of radiation sources, said method comprising: providing Jdifferent electromagnetic sources of radiation, each source of said Jsources characterized by a different function of wavelength and angulardistribution of its emitted radiation, said J≧2; providing thesubstrate, said substrate comprising a base layer and I stacks on thebase layer, said I≧2, wherein P_(j) denotes a same normally incidentenergy flux on each stack from source j such that P_(j) is specific tosource j for j=1, 2, . . . , J; concurrently exposing the I stacks toradiation from only one source t(i) of the J sources in each exposurestep i of I independent exposure steps such that either a firstcondition or a second condition is satisfied; wherein the firstcondition is that said only one source t(i) is selected from the Jsources in exposure step i such that an actual energy flux V_(i)transmitted into the substrate via stack i in exposure step i is maximalthrough deployment of said only one source t(i) as compared withdeployment of any remaining source of the J sources for i=1, 2, . . . ,and I, wherein an error E being a function of |V₁−S₁|, |V₂−S₂|, . . . ,|V_(I)−S_(I)| is about minimized with respect to P_(i) for i=1, 2, . . ., I, wherein S_(i) denotes a specified target energy flux targeted to betransmitted via stack i into the substrate such that S_(i) is specificto each stack i for i=1, 2, . . . , I, wherein J≦I; wherein the secondcondition is a specified design condition on the substrate pertaining toa device parameter of the substrate.
 12. The method of claim 11, whereinE=Σ_(i)(V_(i)−S_(i))², wherein Σ_(i) denotes a summation over i from i=1to i=I.
 13. The method of claim 11, wherein T_(ij) is the fraction ofthe energy flux P_(j) that is transmitted via stack i into thesubstrate, and wherein V_(i)=T_(i,t(i))P_(t(i)), for i=1, 2, . . . , I.14. The method of claim 11, wherein the first condition is satisfied.15. The method of claim 11, wherein the second condition is satisfied.16. The method of claim 11, wherein J=I.
 17. The method of claim 11,wherein J<I.
 18. The method of claim 11, wherein a first source of the Jsources is monochromatic.
 19. The method of claim 11, wherein a firstsource of the J sources is polychromatic.
 20. The method of claim 11,wherein the electromagnetic radiation from a first source of the Jsources is unidirectional and is normally incident upon the I stacks.21. The method of claim 11, wherein the electromagnetic radiation from afirst source of the J sources is non-normally incident upon the Istacks.
 22. A system for serially irradiating a substrate by a pluralityof radiation sources, said substrate comprising a base layer and Istacks on the base layer, said system comprising: J differentelectromagnetic sources of radiation, each source of said J sourcescharacterized by a different function of wavelength and angulardistribution of its emitted radiation, said J≧2; means for concurrentlyexposing the I stacks to radiation from only one source t(i) of the Jsources in each exposure step i of I independent exposure steps suchthat either a first condition or a second condition is satisfied,wherein I≧2, and wherein P_(j) denotes a same normally incident energyflux on each stack from source j such that P_(j) is specific to source jfor j=1, 2, . . . , J; wherein the first condition is that said only onesource t(i) is selected from the J sources in exposure step i such thatan actual energy flux V_(i) transmitted into the substrate via stack iin exposure step i is maximal through deployment of said only one sourcet(i) as compared with deployment of any remaining source of the Jsources for i=1, 2, . . . , and I, wherein an error E being a functionof |V₁−S₁|, |V₂−S₂|, . . . , |V_(I)−S_(I)| is about minimized withrespect to P_(i) for i=1, 2, . . . , I, wherein S_(i) denotes aspecified target energy flux targeted to be transmitted via each stack iinto the substrate such that S_(i) is specific to each stack i for i=1,2, . . . , I, wherein J≦I; and wherein the second condition is aspecified design condition on the substrate pertaining to a deviceparameter of the substrate.
 23. The method of claim 22, whereinE=Σ_(i)(V_(i)−S_(i))², wherein Σ_(i) denotes a summation over i from i=1to i=I.
 24. The method of claim 22, wherein T_(ij) is the fraction ofthe energy flux P_(j) that is transmitted via stack i into thesubstrate, and wherein V_(i)=T_(i,t(i))P_(t(i)), for i=1, 2, . . . , I.25. The system of claim 22, wherein the first condition is satisfied.26. The system of claim 22, wherein the second condition is satisfied.